The Premier League champions were outplayed for large periods by their relegation-threatened FA Cup final opponents, but snatched victory with a brilliant late Carlos Tevez goal.
Mancini said: "We worked really hard because Wigan played really well and every game we play against them is difficult because they play football and they like to win every game.
"I don't think we played very well because we were so tired. Only in the last 20-25 minutes we played better.
"They played very well, they don't deserve to be relegated."
Mancini felt his players were probably still feeling the effects of their energy-sapping FA Cup semi-final win over Chelsea at Wembley on Sunday.
When asked if there may have been a hangover, he said: "Probably yes, only three days.
"For this reason we played with four, five new players. We spent a lot of energy against Chelsea was difficult."